Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations

2008 ◽  
Vol 20 (4) ◽  
pp. 1065-1090 ◽  
Author(s):  
Wenlian Lu ◽  
Tianping Chen
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Network Models ◽  
Almost Periodic ◽  
Filippov Solution ◽  
Neural Network Models ◽  
Delayed Neural Networks ◽  
Special Cases ◽  
Discontinuous Activations

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.

scholarly journals Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis

2021 ◽  
Vol 5 (3) ◽  
pp. 78
Author(s):  
Ivanka Stamova ◽  
Sotir Sotirov ◽  
Evdokia Sotirova ◽  
Gani Stamov
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Fractional Derivatives ◽  
Network Models ◽  
Almost Periodicity ◽  
Periodic Waves ◽  
Almost Periodic ◽  
Neural Network Models ◽  
Caputo Fractional Derivatives ◽  
Periodicity Analysis

In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Leffler stability notion for the almost periodic solution is defined and studied. In addition, a robust global perfect Mittag–Leffler stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models.

scholarly journals Synchronization for an Array of Coupled Cohen-Grossberg Neural Networks with Time-Varying Delay

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Haitao Zhang ◽  
Tao Li ◽  
Shumin Fei
Keyword(s):  
Neural Networks ◽  
Convex Combination ◽  
Network Models ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Neural Network Models ◽  
Time Varying Delay ◽  
Weighting Matrix ◽  
Special Cases ◽  
Varying Delay

This paper makes some great attempts to investigate the global exponential synchronization for arrays of coupled delayed Cohen-Grossberg neural networks with both delayed coupling and one single delayed one. By resorting to free-weighting matrix and Kronecker product techniques, two novel synchronization criteria via linear matrix inequalities (LMIs) are presented based on convex combination, in which these conditions are heavily dependent on the bounds of both the delay and its derivative. Owing to that the addressed system can include some famous neural network models as the special cases, the proposed methods can extend and improve those earlier reported ones. The efficiency and applicability of the presented conditions can be demonstrated by two numerical examples with simulations.

scholarly journals On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

2021 ◽  
Vol 38 (1) ◽  
pp. 67-94
Author(s):  
DAVID CHEBAN ◽  
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Global Attractor ◽  
Difference Equations ◽  
First Integral ◽  
Almost Periodic ◽  
Almost Automorphic ◽  
Stable Solutions ◽  
Autonomous Dynamical Systems

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

scholarly journals Dynamical Analysis of Bayesian Inference Models for the Eriksen Task

2009 ◽  
Vol 21 (6) ◽  
pp. 1520-1553 ◽  
Author(s):  
Yuan Sophie Liu ◽  
Angela Yu ◽  
Philip Holmes
Keyword(s):  
Dynamical Systems ◽  
Network Models ◽  
Neural Substrates ◽  
Dynamical Analysis ◽  
Model Parameters ◽  
Neural Network Models ◽  
Inference Models ◽  
Central Target ◽  
Insight Into ◽  
Discrete Time Dynamical Systems

The Eriksen task is a classical paradigm that explores the effects of competing sensory inputs on response tendencies and the nature of selective attention in controlling these processes. In this task, conflicting flanker stimuli interfere with the processing of a central target, especially on short reaction time trials. This task has been modeled by neural networks and more recently by a normative Bayesian account. Here, we analyze the dynamics of the Bayesian models, which are nonlinear, coupled discrete time dynamical systems, by considering simplified, approximate systems that are linear and decoupled. Analytical solutions of these allow us to describe how posterior probabilities and psychometric functions depend on model parameters. We compare our results with numerical simulations of the original models and derive fits to experimental data, showing that agreements are rather good. We also investigate continuum limits of these simplified dynamical systems and demonstrate that Bayesian updating is closely related to a drift-diffusion process, whose implementation in neural network models has been extensively studied. This provides insight into how neural substrates can implement Bayesian computations.

scholarly journals RANDOM ATTRACTORS OF STOCHASTIC LATTICE DYNAMICAL SYSTEMS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350041 ◽  
Author(s):  
ANHUI GU
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Existence And Uniqueness ◽  
Random Attractor ◽  
Hurst Parameter ◽  
Random Dynamical System ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Random Attractors ◽  
Lattice Dynamical Systems

This paper is devoted to consider stochastic lattice dynamical systems (SLDS) driven by fractional Brownian motions with Hurst parameter bigger than 1/2. Under usual dissipativity conditions these SLDS are shown to generate a random dynamical system for which the existence and uniqueness of a random attractor are established. Furthermore, the random attractor is, in fact, a singleton sets random attractor.

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